Möbius strip

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The MÃÆ' ¶bius strip or MÃÆ' ¶bius band ( UK: , US: ; German: ['mÃÆ'¸: bi ?? s] ), also spelled Mobius or Moebius , is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one limit. The MÃÆ' ¶bius strip has a non-porous mathematical property. This can be realized as a ruled surface. It was discovered independently by German mathematicians August Ferdinand MÃÆ'¶bius and Johann Benedict Listing in 1858.

An example of a MÃÆ' ¶bius strip can be made by taking a strip of paper and giving it a half-touch, and then joining the end of the strip to form a circle. However, the MÃÆ' ¶bius strip is not just the surface of a precise size and shape, like a half-rolled piece of paper depicted in the illustration. In contrast, mathematicians refer to the closed Mábius band as a homeomorphic surface to this strip. The limit is a simple closed curve, that is, homeomorphic to the circle. This allows for a very wide variation of the geometrical version of the MÃÆ'¶bius band as surfaces that each have a definite size and shape. For example, each rectangle can be attached to itself (by identifying one side with opposite edges after orientation reversal) to create a MÃÆ'¶bius band. Some of them can be easily modeled in Euclidean space, and others can not.

A half-clockwise rotation gives the embedding a different MÃÆ' ¶bius strip of half-turn counterclockwise - that is, as an object embedded in the Euclidean space, the MÃÆ' ¶bius strip is a chiral object with right or left-handedness. However, the underlying topological space in the MÃÆ' ¶bius strip is homeomorphic in each case. Unlimited number of different topological embryos from the same topological space into the three-dimensional space exists, since the MÃÆ' ¶bius strip can also be formed by rotating the odd strip several times larger than one, or by cutting and rotating the strips, before joining the the tip. The complete open MÃÆ' ¶bius band is an example of a topological surface that is closely related to the standard MÃÆ' ¶bius strip, but it is not homeomorphic to it.

Finding an algebraic equation, a solution that has a MÃÆ' ¶bius strip topology, is very easy, but, in general, this equation does not describe the same geometric shape derived from the bent paper model described above. In particular, the rotating paper model is a surface that can be developed, has a Gaussian zero curvature. A system of differential-algebraic equations describing this type of model was published in 2007 along with its numerical solution.

The Euler characteristic of the MÃÆ' ¶bius strip is zero.

Video Möbius strip

Properti

The MÃÆ' ¶bius Strip has some curious properties. The lines drawn from the stitches in the middle reunite at the seams, but on the other side. If resumed, the line meets the starting point, and is twice longer than the original strip. This single continuous curve shows that the MÃÆ' ¶bius strip has only one boundary.

Cutting the MÃÆ' ¶bius strip along the center line with scissors produces a long lane with two full twists in it, not two separate lanes; the result is not a strip of MÃÆ' ¶bius. This happens because the original strip has only one side that is twice as long as the original strip. Cutting creates a second independent edge, half of which is on each side of the scissors. Cutting a new, longer strip, in the middle makes two twisted turns, each with two full twists.

If the strip is cut about a third of the way from the edge, it creates two strips: One is a thinner MÃÆ'¶bius strip - it is the middle third of the original strip, which consists of one-third of the width and the same length with the original strip. The other is a longer but thin strip with two full twists in it - this is the original edge edge environment, and it is made up of one-third of the width and twice the original strip length.

Other analog strips can be obtained by combining strips with two or more half-twists in them instead of one. For example, a strip with three and a half bends, when divided lengthwise, into a bent strip tied in a trefoil node. (If the node is unraveled, the strip has eight and a half bends.) Strips with N half bend, when split, into strips with N 1 full bend. Giving an extra lap and reconnecting the tip produces a number called a paradromic ring.

Maps Möbius strip

Geometry and topology

Salah satu cara untuk merepresentasikan strip MÃÆ' ¶bius sebagai bagian dari ruang Euclidean tiga dimensi menggunakan parametrization:

${\ displaystyle x (u, v) = \ kiri (1 {\ frac {v} {2}} \ cos {\ frac {u} {2}} \ kanan) \ cos u}  ÂÂ$

${\ displaystyle y (u, v) = \ kiri (1 {\ frac {v} {2}} \ cos {\ frac {u} {2}} \ kanan) \ sin u}  ÂÂ$

${\ displaystyle z (u, v) = {\ frac {v} {2}} \ sin {\ frac {u} {2}}}  ÂÂ$

where 0 <= u & lt; 2? and -1 <= v <= 1 . This creates a width MÃÆ' ¶bius strip whose center circle has a radius of 1, located in the xy field and centered on (0, 0, 0) . The parameters u run around the dash while v move from one side to the other.

Dalam koordinat kutub silindris ( r , ? , z ) , versi tak terbatas dari strip MÃÆ' ¶bius dapat diwakili oleh persamaan:

$            {\displaystyle         }$ log                   (          r         )          sin                              (                                                         1                  2                                         ?                      )                   =          z          cos                              (                                                         1                  2                                         ?                      )                  .                  {\ displaystyle \ log (r) \ sin \ left ({\ frac {1} {2}} \ theta \ right) = z \ cos \ kiri ({\ frac {1} {2}} \ theta \ right).}   

Penandaan isometrik terlebar dalam 3-ruang

If the fine MÃÆ' ¶bius strip in three spaces is rectangular - that is, made from identifying two opposite sides of the geometric rectangle with bending but not stretching the surface - hence such embedding is known to be possible if the aspect ratio of the rectangle is greater than the root the square of three. (Note the shorter side of the identified rectangle to get the MÃÆ' ¶bius strip.) For aspect ratio less than or equal to the square root of the three, however, the fine insertion of the rectangular MÃÆ' ¶bius strip into three spaces may not maybe.

When aspect ratio approaches limiting ratio ? 3 from above, each strip of such a rectangular MÃÆ' ¶bius in three spaces seems to be close to a form that within the boundary can be thought of as a strip of three equilateral triangles, folded on top of each other so they occupy only one equilateral triangle in three spaces.

If the MÃÆ' ¶bius strips in three spaces are only once differentially differentiated continuously (in symbol: C ¹ ), however, the Nash-Kuiper theorem shows that there is no lower threshold.

The method of making MÃÆ' ¶bius strips of rectangular strips too wide for merely rotating and joining (eg, rectangles of only one unit of length and one unit width) is first folded back and forth using a number of even folds. - "accordion folds" - so folds folded into narrow enough that they can be twisted and spliced, as long as one long enough strip can be joined. With two folds, for example, the 1 ÃÆ'â € "1 strip will be a 1 ÃÆ'â €" 1/3 cross-shaped strip of 'N' and will remain 'N' after half a round. This folded strip, three times along the width, will be long enough to then join at the end. This method works in principle, but becomes impractical after quite a lot of folds, if paper is used. Using plain paper, this construction can be folded flat, with all layers of paper in one field, but mathematically, whether this is possible without stretching the rectangular surface is unclear.

Topology

Topologically, the strip MÃÆ' ¶bius can be defined as a square [0, 1] ÃÆ'â € "[0, 1] with the top and bottom side identified by the relation ( x , 0) ~ (1 - x , 1) for 0 <= x <= 1, as in the diagram on the right.

The less used MÃÆ' ¶bius strip presentation is the result of the torus topology. Torus can be constructed as square [0, 1] ÃÆ'â € "[0, 1] with the edge identified as (0, y ) ~ (1, < i> y ) (glue left to right) and ( x , 0) ~ ( x , 1) (bottom glue). If one then also identifies ( x , y ) ~ ( y , x ) , then someone gets a strip of MÃÆ' ¶bius. The quadratic diagonal (the ( x , x ) points in which both coordinates agree) becomes the boundary of the MÃÆ' ¶bius strip, and carries the orbifold structure , which geometrically corresponds to "reflection" - geodesic (straight line) on the MÃÆ' ¶bius strip reflects back edges to the strip. Notationally, it is written as T ² /S ₂ - 2-torus given by the group action of the symmetric group on two letters (switching coordinates), and it can be considered as space configuration of two irregular points on the circle, may be the same (edges corresponding to the same points), with torus corresponding to the two dots ordered on the circle.

The MÃÆ'¶bius Strip is a compact two-dimensional manifold (ie surface) with a limit. This is a standard example of a non-oriented surface. In fact, the MÃÆ' ¶bius strip is the epitome of a non-orientation topology phenomenon. This is because the two-dimensional (surface) shape is the lowest dimensional form that is impossible to model and the MÃÆ'¶bius strip is the only only surface which is a subspace of any non-porous surface. Consequently, any surface is unreliable if and only if it contains MÃÆ'¶bius tape as a subspace.

The MÃÆ' ¶bius Strip is also a standard example used to describe the mathematical concepts of fiber bundles. Specifically, it is a nontrivial bundle of the S ¹ circle with the fiber unit interval, I = [0, 1] . Just look at the edge of the MÃÆ' ¶bius strip giving the two-point trio number (or Z ₂ ) above S ¹ .

Computer graphics

The simple construction of the MÃÆ' ¶bius strip that can be used to describe it in a computer graph or modeling package is:

• Take a rectangular strip. Rotate around fixed point not in the field. At each step, it also rotates the strip along the line in its plane (the line dividing the strip into two) and perpendicular to the radius of the main orbital. The surface produced in one complete revolution is the MÃÆ'¶bius strip.
• Take the MÃÆ' ¶bius strip and cut it along the center of the strip. It forms a new strip, which is a joined rectangle by rotating one end of the whole turn. By cutting it to the center again, it forms two intact strips that are interconnected.

Open the band MÃÆ'¶bius

The open MÃÆ'¶bius band is formed by removing the standard MÃÆ'¶bius band limits. It's built from sets S = {( x , y )? R ² Ã,: 0 <= x <= 1 and 0 & lt; y & lt; 1} by identifying (pasting) the points (0, y ) and (1, 1 - y ) for all 0 & lt; y & lt; 1 .

It can be constructed as a constant positive, negative, or zero (Gaussian) constant curvature surface. In the case of negative and zero curvature, the MÃÆ'¶bius band can be constructed as a complete surface (geodesic), which means that all geodesies ("straight lines" on the surface) can be extended indefinitely in both directions.

Constant negative curvature: As with airplanes and open cylinders, the MÃÆ'¶bius band is open to recognizing not only a complete metric of curvature of 0, but also a complete metric of constant negative curvature, say -1. One way to see this is to start with a half-plane model (PoincarÃÆ'Â ©) from the hyperbolic field H, ie H = {( x , y ) ? R ² | y & gt; 0} with Riemannian metrics provided by ( dx ² )/ y ² . Isometric orientation preservation of this metric is all the map f : H -> H of the form f ( z ) Ã,: = ( az b )/( cz d ) , where a , b , c , d is a real number that satisfies > - bc = 1 . Here z is a complex number with Im ( z ) & gt; 0 , and we have identified H with { z ? C | Im ( z ) & gt; 0} is endowed with the mentioned Riemannian metrics. Then an orientation-reverse isometry g of H is given by g ( z ) Ã,: = -conj ( z ) , where the conjunction ( z ) shows the complex conjugation z . These facts imply that mapping h Ã,: H -> H is provided by h ( z ) Ã,: = -2? Conj ( z ) is the orientation-reverse isometry H which results in an infinite cyclic group G isometry. (The box is isometric h ( z ) Ã,: = 4? Z , which can be declared as ( 2z 0 )/( 0z 1/2 ) .) of this group action can be easily seen as the topology of the MÃÆ'¶bius band. But it is also easy to verify that it is complete and not compact, with a constant negative curvature of -1.

The isometric group of the MÃÆ'¶bius band is 1-dimensional and isomorphic to the orthogonal group SO (2).

(Constant) zero curvature: This can also be constructed as a complete surface, by starting with the R ² field defined by 0 <= y <= 1 and identify ( x , 0) with (- x , 1) for all x in R (real). The resulting metric makes the MÃÆ'¶bius band open into a complete (geodetic) flat surface (ie, having a Gaussian curvature equal to 0 everywhere). This is a metric only on the band MÃÆ'¶bius, up to a uniform scale, which is flat and complete.

The isometric group of the MÃÆ'¶bius band is 1-dimensional and isomorphic in the orthogonal group SO (2).

Constant positive curvature: A positive constant curvature MÃÆ'¶bius band can not be complete, because it is known that the only complete surface of constant positive curvature is a sphere and a projective field. The projection field P ² of the constant curvature 1 can be constructed as a result for the soccer unit S ² in R ³ by antipodal map A : S ² -> S ² , defined by A ( x , y , z ) = (- x , - y , - z ) . The band MÃÆ' ¶bius opens homomorphically to the projective field once pierced, ie, P ² with a single point deleted. It may be considered the closest that the MÃÆ'¶bius band of constant positive curvature can be a complete surface: just one point.

The isometric groups of the MÃÆ'¶bius band are also 1-dimensional and isomorphic to the orthogonal group O (2).

The non-oriented strip space in the plane is diffeomorphic to the open Mábius band. To see why, let L (? ) show the line through the origin point at the angle ? to the positive x-axis. For each L (? ) there is a family P (? ) of all lines in a plane perpendicular to < i> L (? ). In topology, the family P (? ) is just a line (since each row in P (? ) L (? ) in just one dot). In this way, like ? increases in range 0Ã, Â ° <= ? & lt; 180 °, the line L (? ) represents a different line of lines on the plane. But when ? reaches 180 °, L (180 °) is identical to L (0), and once the family P (0 Â °) and P (180 °) straight lines are also identical families. The L (0 °) line, however, has returned to itself as L (180 °) pointing in the opposite direction . Each line in the plane corresponds to a row in some families P (? ), to exactly one ? , for 0Ã, Â ° <= ? & lt; 180 °, and P (180 °) is identical to P (0 °) but again pointing in the opposite direction. This ensures that the space of all lines on the plane - the merging of all L (? ) for 0Ã, Â ° <= ? <= 180Ã, Â ° - is the open MÃÆ'¶bius band.

Linear linear transformation group GL (2, R ) from the plane to itself (real matrix 2 ÃÆ'â € "2 with non-zero determiners) naturally induces a bias of line space in a plane on itself, which forms a self-homomorph group of line spaces. Therefore the same group formed the self-homeomorphisms group of the MÃÆ'¶bius band described in the preceding paragraph. But there is no metric in the line space on the plane that does not change under the action of this homeomorphism group. In this sense, the line space on the plane does not have any natural metrics on it.

This means that the MÃÆ'¶bius band has a natural 4-dimensional Lie self-homeomorphisms group, given by GL (2, R ) , but this high symmetry level can not exhibited as any isometric metric group.

Band MÃÆ'¶bius with round boundary

The edge, or boundary, of the MÃÆ' ¶bius strip is homeomorphic (topologically equivalent) to the circle. Under the usual embedding of the strips in the Euclidean space, as above, the boundary is not a true circle. However, it is possible to embed the MÃÆ' ¶bius strip in three dimensions so that the limit is a perfect circle located in several fields. For example, see Figures 307, 308, and 309 "Geometry and Imagination".

A much more geometric embedding begins with a minimal Klein bottle dipped in 3-ball, as found by Blaine Lawson. We then took half of this Klein bottle to get the MÃÆ'¶bius band embedded in a 3-ball (ball unit in 4-chamber). The result is sometimes called "Sudanese MÃÆ'¶bius Band", in which "sudanese" does not refer to the Sudanese state but to the names of the two topologists, Sue Goodman and Daniel Asimov. Applying stereographic projections to Sudanese bands puts them in 3-dimensional space, as can be seen below - a version because George Francis can be found here.

Dari botol Klein minimal Lawson, kami mendapatkan penyematan band ke dalam 3-bola S ³ , yang dianggap sebagai bagian dari C ² , yang secara geometris sama dengan R ⁴ . Kami memetakan sudut ? , ? ke bilangan kompleks z ₁ , z ₂ via

${\ displaystyle z_ {1} = \ sin \ eta \, e ^ {i \ varphi}}  ÂÂ$

${\ displaystyle z_ {2} = \ cos \ eta \, e ^ {i \ varphi/2}.}  ÂÂ$

Here parameter ? runs from 0 to ? and ? runs from 0 to 2 ? . Since | z ₁ | ² | z ₂ | ² = 1 , the fully embedded surface is at S ³ . The strip limit is provided by | z ₂ | = 1 (according to ? = 0, ? ), which is clearly a 3-ball circle.

To get a MÃÆ' ¶bius embedding strip in R ³ a map S ³ b> ³ via stereographic projection. The projection point can be any point on S ³ which is not located on the embedded MÃÆ'¶bius strip (this excludes all common projection points). One possible option is $Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â1Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â/Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\sqrt{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}$,             me             /                                                       2                                               }                   Annotation encoding = "application/x-tex"> {\ displaystyle \ left \ {1/{\ sqrt {2}}, i/{\ sqrt {2}} \ right \}}   . The stereographic projection maps the circle to the circle and maintains the circular boundary of the strip. The result is a smooth insertion of the MÃÆ'¶bius strip to R ³ with a circular edge and no self-intersection.

The Sudanese band MÃÆ'¶bius in three geological spherical bundles is a fiber bundle over a large circle, whose fibers are semicircular large. The most symmetrical image of the band's stereographic projection to R ³ is obtained by using a projection point located on a large circle running through the midpoint of each semicircle. Each option like a projection point produces an image that matches the other. But since such a projection point lies in the band MÃÆ'¶bius itself, the two image aspects are significantly different from the case (pictured above) where the point is not on the band: 1) the image at R ³ is not a full MÃÆ'¶bius band, but bands with a single point are removed (from the center line); and 2) the image is unrestricted - and as it is further away from the origin R ³ , it is getting closer to an airplane. But this stereographic image version has a group of 4 symmetries in R ³ (isomorphic with the Klein 4 cluster), compared to the limited version illustrated above having a unique group of group 2 symmetry groups (If all symmetries and not just the isometric-preservation orientation of R ³ are allowed, the number of symmetries in each case is double.)

But the most geometrically symmetrical version of all is the original Sudanese Mábius band in a three-sided sphere, where the complete symmetry group is isomorphic to the Lie O group (2). Having an infinite cardinality (which is from the continuum), this is much larger than the symmetry group of any possible embedding of the MÃÆ'¶bius band in R ³ .

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Related objects

The closely related 'strange' geometric object is the Klein bottle. The Klein Bottle can be produced by attaching two MÃÆ' ¶bius strips along the edges; this can not be done in the usual Euclidean three-dimensional space without making self-intersections.

Another closely related manifold is a real projective field. If the round disk is cut from the real projective field, what remains is the MÃÆ' ¶bius strip. Go the other way, if someone puts the dish to the MÃÆ' ¶bius strip by identifying their boundaries, the result is a projective field. To visualize this, it would be helpful to damage the MÃÆ'¶bius strips so that the limits are ordinary circles (see above). Real projective fields, such as Klein bottles, can not be embedded in three dimensions without self-intersection.

In graph theory, the MÃÆ'¶bius ladder is a cubic graph that is closely related to the MÃÆ'¶bius strip.

In 1968, Gonzalo VÃÆ' Â © lez Jahn (UCV, Caracas, Venezuela) discovered a three-dimensional body with MÃÆ'¶bian characteristics; this was later described by Martin Gardner as a prismatic ring that became a toroidal polyhedrons in the Mathematical Mathematics column of August 1978 in Scientific American.

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Apps

There are several technical applications for the MÃÆ' ¶bius strip. Giant strip MÃÆ'¶bius has been used as a long-lasting conveyor belt because the entire surface area of ​​the belt gets the same amount of wear, and as a recording of continuous loop (to double play time). The MÃÆ' ¶bius strip is common in manufacturing computer printer fabrics and typewriter ribbons, because they allow the tape to be twice as wide as the print head when using both parts evenly.

The MÃÆ'¶bius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented a similar technology in 1894: "Coil for Electro Magnet" is intended for use with the global wireless power transmission system.

The MÃÆ' ¶bius Strip is an irregular two-point configuration space on a circle. Consequently, in musical theory, the space of all two-note chords, known as dyads, takes the form of the MÃÆ' ¶bius strip; this and generalization for more points is a significant application of orbifolds to music theory.

In physics/electro-technology as:

• A compact resonator with a half resonant frequency of an identically constructed linear coil
• Resistor without induction
• Superconductor with high transition temperature
• MÃÆ'¶bius resonator

In chemistry/nanoscale technology as:

• Molecular node with special characteristics (Knotane [2], Chirality)
• Molecular Machines
• Volume graphene (nano-graphite) with new electronic characteristics, such as helical magnets
• Special aromaticity type: Miguium aromatisitas
• Charged particles caught in the magnetic field of the earth that can move on the MÃÆ'¶bius band
• B1 cyclotida (cyclic protein), plant active ingredient Oldenlandia affinis , contains the MÃÆ'¶bius topology for the peptide backbone.

In stage magic

The principle of the MÃÆ' ¶bius strip has been used as a method of creating magical illusions. The trick, known as the Afghan band, was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such as Harry Blackstone Sr and Thomas Nelson Downs.

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See also

• Cross-cap
• torus Umbilic
• Ribbon theory

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References

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External links

• The MÃÆ'¶bius Gear - A functional planetary gear model in which one tooth is a strip of MÃÆ' ¶bius
• Weisstein, Eric W. "MÃÆ'¶bius Strip". MathWorld .

Source of the article : Wikipedia